Normal elements of completed group algebras over ${\rm SL}_3(\mathbb{Z}_p) $

TL;DR

This paper proves that all nonzero normal elements in the completed group algebra over ${\rm SL}_3(\mathbb{Z}_p)$ are units, resolving an open question and correcting previous errors through computational methods.

Contribution

It establishes that nonzero normal elements in the algebra are units, providing a definitive answer to an open problem and rectifying earlier inaccuracies.

Findings

All nonzero normal elements are units.

Resolved an open question in the field.

Corrected previous mistakes in related research.

Abstract

Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm SL}_3(\mathbb{Z}_p)$ is a unit. This give a positive answer to an open question in \cite{WeiBian2} and make up for an earlier mistake in \cite{WeiBian1} simultaneously.